Question: Find the sum of $997 + 992 + 987 +... + 257 + 252$.
Answer: Getting started We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $5$ less than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {997})$ and the last term $(a_n = {252})$ are given in the question. We need to find $n$ (the number of terms). Step 1: Find $n$ (the number of terms) The sequence decreases by $997 - 252 = 745$ from the first term to the last term. Because the sequence decreases by $5$ each time, it takes $\dfrac{745}{5} = 149$ terms to get from the first term to the last term. We still need to count the first term, so there are $149 + 1 = {150}$ terms in the sequence. In other words, $n = {150}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{150}}&= \dfrac {\left({997} + {252} \right)}{2} \cdot {150} \\\\ S_{{150}} &= 624.5 \left(150\right) \\\\ S_{{150}} &= 93{,}675\end{aligned}$ The answer $ 93{,}675$